#8360912

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,47)(46,48);(2,7)(3,11)(4,14)(13,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(35,36)(37,40)(43,44)(45,47);(2,7)(5,9)(8,10)(13,16)(19,26)(22,24)(27,29)(30,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8268108);